
Indeed, let \(L_{\infty }(\pi _P)\)  be the set of \(\pi \) -almost surely bounded functions, equipped with the infinity-norm \(\Vert f\Vert _{\infty }:=\mbox{ess sup}_{x \in \Omega _n} |f(x)|\) . Then \(L_{\infty }(\pi _P)\)  is a Banach space. In addition, for any \(P \in \mathcal {P}\) , \(\mathbf {P}^* \mathbf {P}\)  is self-adjoint, bounded linear operator on \(L_2(\pi )\) . Define the operator \(\mathbf {\pi }\)  on \(L_2(\pi )\)  as \(\mathbf {\pi }(f)(x):=\pi (f)\) . This is also a self-adjoint bounded operator. Let \(\mathbf {M}:=(\mathbf {P}^*)^{\tau _P(\varepsilon )}\mathbf {P}^{\tau _P(\varepsilon )}-\mathbf {\pi }\) . Then, we can express the absolute spectral gap of \(\gamma _P^*\)  of \((\mathbf {P}^*)^{\tau _P(\varepsilon )}\mathbf {P}^{\tau _P(\varepsilon )}\)  as [1]}:
\(\gamma _P^*=1-\sup \bigg \lbrace |\lambda |: \lambda \in \mathcal {S}_2(\mathbf {M})\bigg \rbrace  ,\) 
